Projective Isomonodromy and Galois Groups

等单性与Painlevé型方程，案例研究

Isomonodromy and Painlevé Type Equations, Case Studies

本文研究由至多含两个奇点的线性ODE等单性族诱导的Painlevé型方程，分类案例并发现新层级及$P_1$的伴随方程。

Galois groups for integrable and projectively integrable linear difference equations

We consider first-order linear difference systems over $\mathbb{C}(x)$, with respect to a difference operator $σ$ that is either a shift $σ:x\mapsto x+1$, $q$-dilation $σ:x\mapsto qx$ with $q\in{\mathbb{C}^\times}$ not a root of unity, or Mahler operator $σ:x\mapsto x^q$ with $q\in\mathbb{Z}_{\geq 2}$. Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable "after moding out by scalars." We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or pro…